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A001682 Numbers k such that 3^k, 3^(k+1) and 3^(k+2) have the same number of digits.
(Formerly M5109 N2213)
3

%I M5109 N2213 #36 Feb 11 2021 23:00:33

%S 0,21,42,65,86,109,130,151,174,195,218,239,262,283,304,327,348,371,

%T 392,415,436,457,480,501,524,545,568,589,610,633,654,677,698,721,742,

%U 763,786,807,830,851,874,895,916,939,960,983,1004,1027,1048

%N Numbers k such that 3^k, 3^(k+1) and 3^(k+2) have the same number of digits.

%C Equivalently the fractional part of n*log(3) lies between 0 and 1 - 2*log(3), about 0.04576; 1 - 2*log(3) is also the density of the sequence. - Kevin Costello, Aug 08 2002

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Reinhard Zumkeller, <a href="/A001682/b001682.txt">Table of n, a(n) for n = 1..1000</a>

%H Murray Klamkin and Joe Lipman, <a href="https://www.jstor.org/stable/2309610">Problem E1238</a>, Amer. Math. Monthly, 64 (1957), 367.

%t Select[Range[0, 2000], IntegerLength[3^#] == IntegerLength[3^(#+1)] == IntegerLength[3^(#+2)]&] (* _Jean-François Alcover_, Nov 24 2011 *)

%t Flatten[Position[Partition[IntegerLength[3^Range[0,1100]],3,1],_?( Length[ Union[#]]==1&),{1},Heads->False]]-1 (* _Harvey P. Dale_, Jan 31 2015 *)

%t SequencePosition[IntegerLength[3^Range[0,1200]],{x_,x_,x_}][[All,1]]-1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Dec 12 2018 *)

%o (Haskell)

%o a001682 n = a001682_list !! (n-1)

%o a001682_list = [k | k <- [0..], let m = 3^k, a055642 m == a055642 (9*m)]

%o -- _Reinhard Zumkeller_, Oct 10 2011

%Y First differences give A151910.

%Y Cf. A055642, A000244.

%K nonn,base,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _R. K. Guy_ and _Emeric Deutsch_, Mar 09 2005

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)